The Vertex of a Parabola

The
vertex
of a
parabola
is the point where the parabola crosses its axis of symmetry. If the coefficient of the
x
2
term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “
U
”-shape. If the coefficient of the
x
2
term is negative, the vertex will be the highest point on the graph, the point at the top of the “
U
”-shape.

The standard equation of a parabola is

y
=
a
x
2
+
b
x
+
c
.

But the equation for a parabola can also be written in "vertex form":

y
=
a
(
x
−
h
)
2
+
k

In this equation, the vertex of the parabola is the point
(
h
,
k
)
.

You can see how this relates to the standard equation by multiplying it out:

y
=
a
(
x
−
h
)
(
x
−
h
)
+
k
y
=
a
x
2
−
2
a
h
x
+
a
h
2
+
k
.

This means that in the standard form,
y
=
a
x
2
+
b
x
+
c
, the expression
−
b
2
a
gives the
x
-coordinate of the vertex.

Example:

Find the vertex of the parabola.

y
=
3
x
2
+
12
x
−
12

Here,
a
=
3
and
b
=
12
. So, the
x
-coordinate of the vertex is:

−
12
2
(
3
)
=
−
2

Substituting in the original equation to get the
y
-coordinate, we get: